from gapped phases of matter to topological quantum field...
TRANSCRIPT
From gapped phases of matter to TopologicalQuantum Field Theory and back again
Anton Kapustin
California Institute of Technology
June 22, 2020
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 1 / 31
Outline
Gapped phases of matter
Invertible TQFT and invertible phases of matter
Berry-Wess-Zumino-Witten cohomology classes for families of gappedlattice models (based on work with Lev Spodyneiko,arXiv:2001.03454)
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 2 / 31
States of matter
In elementary school, we learn about qualitatively different ”states ofmatter”: solid, liquid and gas.
Solids are crystals on a microscopic scale. The distinction between solidsand other states of matter is based on symmetry:
Liquids and gases are invariant under arbitrary translations androtations of the Euclidean space R3.
Crystals are invariant under some infinite discrete subgroup of thesymmetry group of R3.
There are also states of matter (”liquid crystals”) which are invariantunder arbitrary translations, but not under arbitrary rotations. Liquidcrystal displays (LCD) can be found in many TVs and computer monitors.
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 3 / 31
Phases of matter
Is there some qualitative distinction between liquid and gas? No: one can”deform” one into another by varying both temperature and pressure.
Allowed deformations are those which preserve a natural property:correlations between physical quantities at points p and p′ becomeexponentially small when |p − p′| is large:∣∣〈A(p)B(p′)〉 − 〈A(p)〉〈B(p′)〉
∣∣ ≤ Ce−|p−p′|/ξ.
Loci in the parameter space where this fails are called phase transitions.After removing these loci, the parameter space may fall into severaldisconnected components. These are called phases of matter.
Liquid and gas belong to the same phase of matter. Crystals with differentsymmetry groups are examples of distinct phases of matter.
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 4 / 31
Quantum systems
Traditionally, physicists studied phases at positive temperature. Morerecently, they turned to phases at zero temperature.
A quantum system is described by a triple (A , τt , ψ):
A is a C ∗ algebra
τt is a 1-parameter family of automorphisms of A
ψ is a state on A invariant under τt
ψ gives rise to a representation of A by bounded operators in a Hilbertspace V , so that ψ(A) = 〈0|A|0〉 for some cyclic vector |0〉 ∈ V .
τt gives rise to a Hamiltonian H (unbounded self-adjoint operator on V )which annihilates |0〉.
The ”zero-temperature” condition is: ψ is pure and H ≥ 0. For infinitevolume systems one usually assumes |0〉 is the only vector annihilated byH. Then ψ is called a ground state for τt .
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 5 / 31
Mind the gap!
If 0 is an isolated eigenvalue of H, the system is said to be in a gappedphase. Two systems are in the same gapped phase if they can be deformedinto each other without ”closing the gap”.
Typically, it is assumed that A is an infinite tensor product ⊗p∈ΛAp,where Λ is an infinite discrete subset of Rd , and Ap is a matrix algebra.A ∈ A is a local observable localized on a finite subset X ⊂ Λ ifA ∈ AX = ⊗p∈XAp.
Under some assumptions about the Hamiltonian H (”exponentiallydecaying interactions”) a gap in the spectrum of H implies that for anyfinite subsets X and Y and any A ∈ AX and B ∈ AY one has
|〈AB〉 − 〈A〉〈B〉| ≤ ||A|| · ||B|| · |X | · |Y |e−dist(X ,Y )/ξ.
Thus the notion of a ”gapped phase” is compatible with the notion of aphase based on correlation decay.
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 6 / 31
Examples of gapped phases
Trivial phase (deformation class of a gapped system whereH =
∑p Hp with Hp ∈ Ap, and the state ψ is ”factorized”, i.e.
ψ(AB) = ψ(A)ψ(B) if A ∈ Ap, B ∈ Aq and p 6= q.)
Fractional Quantum Hall phases of 2d insulating systems in amagnetic field
Integer Quantum Hall phases of 2d insulating systems in a magneticfield
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 7 / 31
Fractional Quantum Hall phases
Hall effect at zero temperature with rationally ”quantized” Hallconductance
Localized excitations with fractional charge! (For example, chargee/3, where e is the charge of the electron)
Enthropic Hall effect (flow of entropy perpendicular to temperaturegradient in the limit T → 0).
Localized excitations with ”fractional statistics”! (local excitationscannot be classified as bosons or fermions)
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 8 / 31
Integer Quantum Hall phases
Integrally quantized Hall conductance at zero temperature
Enthropic Hall effect (flow of entropy perpendicular to temperaturegradient in the limit T → 0).
IQHE phases can be distinguished from the trivial phase only by theirresponse to external ”fields” (voltage and temperature).
Both FQHE and IQHE phases have U(1) symmetry, but remain non-trivialeven if one allows deformations which break this symmetry.
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 9 / 31
Topological Quantum Field Theory
It was realized in the late 1980s - early 1990s that both FQHE and IQHEcan be described by Chern-Simons field theory.
Chern-Simons field theory is an example of a 3d TQFT.Can one use TQFT in d + 1 space-time dimension to describe gappedphases of matter in d spatial dimensions in general?
TQFTs have been axiomatized (Atiyah,..., Bayez, Dolan,...,Lurie).Can one use TQFT to classify gapped phases of matter?
There is actually an infinity of classification problems, labeled by thedimension of space d and the symmetry group G . IQHE phasescorrespond to d = 2 and G = U(1).
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 10 / 31
Extended TQFT
Extended TQFT in n space-time dimensions is defined as a symmetricmonoidal functor from one symmetric monoidal (∞, n) category(”source”) to another (”target”).
The ”source” (∞, n) category Bordn has compact 0-manifolds as objects,compact 1-manifolds with boundaries as 1-morphisms, compact2-manifolds with corners as 2-morphisms,..., all the way up to n-manifolds.Higher morphisms are multi-parameter families of n-morphisms.Symmetric monoidal structure arises from disjoint union.
The ”target” (∞, n) category Cn can vary, but its morphisms of degreen − 1 are finite-dimensional vector spaces and morphisms of degree n arelinear maps between them. Symmetric monoidal structures is related totensor product.
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 11 / 31
Invertible TQFTs
It is very difficult to classify TQFTs in general dimensions (although somedimensions might be easier than others). One problem is the absence of anatural choice of a target (∞, n)-category.
An interesting class of TQFTs is invertible TQFTs. TQFTs form amonoid, invertible TQFTs are those which have an inverse.
If all morphisms in an ∞-category are invertible, we get an ∞-groupoidwhich can be replaced with an ∞-loop space (”loop spectrum”).
This is a sequence of pointed spaces X0,X1, . . . , and homotopyequivalences
ΩXn ∼ Xn+1.
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 12 / 31
Classification of invertible TQFTs
Freed, Hopkins, 2016:
For unitary invertible TQFTs, can replace Bordn with the orientedThom spectrum (for spin-TQFTs, spin Thom spectrum)
In the invertible case, there is a ”natural” choice of a target (aparticular loop spectrum).
This leads to a classification of deformation classes of unitary invertibleTQFTs in d + 1 space-time dimensions:
InvBd+1 = π0(Xd+1),
where X0,X1, . . . is the ”Anderson dual” of the oriented Thom spectrum.
For unitary invertible spin-TQFTs:
InvFd+1 = π0(Yd+1),
where Y0,Y1, . . . is the dual of the spin Thom spectrum.
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 13 / 31
Low-dimensional cases
X0 = K (Z, 1), Y0 = K (Z, 1)
X1 = K (Z, 2), Y1 = Z/2× K (Z, 2),
X2 = K (Z, 3), Y2 = Z/2× K (Z/2, 1)× K (Z, 3)
X3 = Z× K (Z, 4), Y3 = Z× E
Freed-Hopkins result implies that the deformation class of a unitaryinvertible TQFTs in n dimensions is determined by its partition functionson all possible closed n-manifolds (oriented or spin).
In the bosonic case, first non-trivial deformation classes appear for d = 2.In the fermionic case, they first appear for d = 0. IQHE phases arebelieved to be described by non-trivial invertible spin-TQFTs in threespace-time dimensions (even if one ignores U(1) symmetry).
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 14 / 31
Gapped phases vs TQFT
For some time if was believed (without much evidence) that any gappedphase can be described by a TQFT.
Recently, intricate examples of gapped systems in 3d have beenconstructed (”fractons”) which are counter-examples to this belief.
The 1-1 relation between TQFTs and gapped phases might still be true ifone restricts to invertible TQFTs and invertible gapped phases.
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 15 / 31
Invertible gapped phases
Gapped lattice systems can be ”stacked” (composed):
(A , τt , ψ), (A ′, τ ′t , ψ′) 7→ (A ⊗A ′, τt ⊗ τ ′t , ψ ⊗ ψ′).
A system (A , τt , ψ) is said to be an invertible phase if it can be stackedwith another system to obtain a system in a trivial phase.
Gapped phases form a commutative monoid, invertible phases form acommutative group.
IQHE phases are invertible. FQHE phases are not.
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 16 / 31
Invertible gapped phases and loop spectra
A. Kitaev proposed to define a Short-Range Entangled phase as aninvertible phase.
Let Wd+1 be the space of d-dimensional systems in an invertible phase(with some fixed symmetry G ). Kitaev gave a heuristic argument that thespaces W1,W2, . . . form a loop spectrum.
Based on some low-dimensional cases, I proposed in 2015 that thisspectrum is related to Thom spectra and that invertible gapped phasescan be described by TQFT.
Freed and Hopkins showed that unitary invertible TQFTs can indeed beclassified using the Anderson dual of Thom spectra.
What about invertible phases?
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 17 / 31
Why it is hard to relate invertible gapped phases andinvertible TQFT
TQFTs can be defined on arbitrary d + 1-dimesional closed manifolds
Lattice models are formulated on Rd (times time) and cannot benaturally defined on a general manifold.
Every bosonic system can be viewed as a fermionic one. Thus there is amap from the set of bosonic phases to the set of fermionic phases.Restricting to invertible ones, get a homomorphism
InvBd+1 → InvFd+1.
On the TQFT side, get a map of spectra X → Y . In particular,π0(X3) = π0(Y3) = Z, and the map is multiplication by 16.
This is related to Rokhlin’s theorem: signature of a spin 4-manifold isdivisible by 16. How would lattice models on R2 know about this?
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 18 / 31
How to ”see” a loop spectrum
Let Wd+1 be the ”space of all invertible system in spatial dimension d”(bosonic, fermionic, or with some additional symmetry). Suppose the
spaces W1,W2, . . . form a loop spectrum. Then
πk(Wd+1) = π0(ΩkWd+1) = π0(Wd+1−k), k ≥ d + 1.
Thus homotopy groups of Wd+1 are predicted in terms of the group ofinvertible phases in lower dimensions.
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 19 / 31
Low-dimensional cases
W1 is homotopy equivalent to the space of projectors with aone-dimensional image, i.e. the space of lines. Thus
W1 ∼ limN→∞
CPN = K (Z, 2).
Therefore we predict:
π1(W2) = π0(W1) = 0, π2(W2) = π1(W1) = 0, π3(W2) = π2(W1) = Z.
Thus the space of 1d lattice systems in a trivial phase must be a K (Z, 3).
In particular, since H3(W2) = Z, to any family of 1d systems in aninvertible phase one should be able to assign a degree-3 cohomology classon the parameter space.
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 20 / 31
Berry curvature and its higher-categorical relatives
Note that H2(W1) = Z. What is the corresponding degree-2 class on theparameter space of a family of 0d invertible lattice systems?
M V. Berry (1984) noticed that to any family of gapped Hamiltonian onecan associate a connection on the line bundle of its ground states (byprojecting the trivial connection on the trivial Hilbert bundle).
The curvature of the Berry connection is a closed 2-form representing the1st Chern class of this line bundle.
For 1d systems, we expect a degree-3 class which we can represent by aclosed 3-form on the parameter space. This would be the Dixmier-Douadyclass of a gerbe.
For 2d systems, expect a 2-gerbe and a closed 4-form, etc.
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 21 / 31
One-dimensional lattice systems
Hamiltonian: H =∑
p∈ZHp, where Hp ∈ A is uniformly bounded andapproximately localized on p:
||[Hp,A]|| = O(|p − q|−∞), ∀A ∈ Aq.
Pure state ψ : A → C gives rise to an irreducible representation of A ona Hilbert space V (via the Gelfand-Naimark-Segal construction) with acyclic vector |0〉. H gives rise to an unbounded self-adjoint operator H onV such that H|0〉 = 0. ψ is called a ground state for H if H ≥ 0.
A system (A ,H, ψ) is called gapped if |0〉 is the unique vector witheigenvalue 0 and the spectrum of H is contained in 0 ∪ [∆,∞) for some∆ > 0.
We are interested in a differentiable family of gapped systems(A ,H(s), ψ(s)), s ∈ M (the parameter space).
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 22 / 31
Looking for closed forms I
For a family of gapped Hamiltonian on a finite-dimensional V , we have aformula for the Berry curvature:
F (2) =i
2
∮dz
2πiTrG (z) dH G (z)2 dH
Here G (z) = 1/(z − H). dF (2) = 0 is checked by a direct computation.
Does not work for systems for d > 0. We can consider
F(2)pq =
i
2
∮dz
2πiTrG (z) dHp G (z)2 dHq.
Although∣∣∣F (2)
pq
∣∣∣ = O(|p − q|−∞), the sum F (2) =∑
pq F(2)pq is still
divergent, thanks to the contribution of points near the diagonal.
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 23 / 31
Looking for closed forms II
Consider instead a well-defined 2-form
F(2)p =
∑q∈Z
F(2)pq .
It is not closed. But is satisfies
dF(2)p =
∑q
F(3)pq ,
where F(3)pq is a 3-form, F
(3)pq = −F (3)
qp , and F(3)pq = O(|p − q|−∞).
In turn F(3)pq satisfies
dF(3)pq =
∑r
F(4)pqr ,
where F(4)pqr is a 4-form which is skew-symmetric in p, q, r , etc.
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 24 / 31
An explicit formula for F(3)pq
F(3)pq =
i
6
∮dz
2πiTr(G 2(z) dH G (z) dHp G (z) dHq
−G (z) dH G (z)2 dHp G (z) dHq
)− (p ↔ q) (1)
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 25 / 31
Vietoris-Rips chain complex
We can formalize this by introducing the Vietoris-Rips (or coarse) chaincomplex.
Let Λ be an infinite subset of Rd which is uniformly discrete and uniformlyfilling.
An n-chain is a skew-symmetric function of n + 1 points on Λ which isbounded and decays rapidly away from the diagonal. Let Cn be the spaceof n-chains, n ≥ 1. Define ∂ : Cn → Cn−1 by
(∂A)p1...pn =∑pn+1
Ap1...pnpn+1 .
Then F(2)p is a 2-form with values in 0-chains, F
(3)pq is a 3-form with values
in 1-chains, etc. Total degree is 2.
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 26 / 31
The descent equation
All the equations we had can be written as
dF (n) = ∂F (n+1).
We will call this the descent equation.
The descent equation was proposed by A. Kitaev in the context of”classical” spin systems on a lattice. We apply it to quantum models.
To construct a closed p-form, we need to contract the p-form-valued(p − 2)-chain F (p) with Vietoris-Rips cochain.
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 27 / 31
Vietoris-Rips cochain complex
An n-cochain is a bounded skew-symmetric function of n + 1 points on Λwhose support is ”co-controlled” (intersection of support with anythickened diagonal is finite). See books by John Roe on coarse geometry.
Let Cn be the space of n-cochains. The differential δ : Cn → Cn+1 is dualto ∂:
〈∂A, α〉 = 〈A, δα〉, ∀A ∈ Cn+1,∀α ∈ Cn.
If α ∈ Cp−2 and δα = 0, then
d〈F (p), α〉 = 〈∂F (p+1), α〉 = 〈F (p+1), δα〉 = 0.
If α = δβ, then 〈F (p), α〉 is exact.
To get a non-trivial degree-p cohomology class on the parameter space,want α to be a closed, but not exact (p − 2)-cochain.
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 28 / 31
Vietoris-Rips cohomology
For Λ = Z ⊂ R, the cohomology is R in degree 1 and zero for all otherdegrees. Degree-1 cohomology is spanned by α(p, q) = f (p)− f (q), wheref : Z→ R is any function which approaches ±1 at ±∞.
Hence we need to let p = 3. Then we get a closed 3-form on theparameter space:
Ω(3) =1
4
∑p,q
F(3)pq (f (p)− f (q)).
Its cohomology class is independent of the choice of f . It is ahigher-categorical version of the cohomology class of the Berry curvature.(Kapustin, Spodyneiko, arXiv:2001.03454)
Similarly, for a family in d dimensions get a closed (d + 2)-form on theparameter space (Wess-Zumino-Witten form).
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 29 / 31
Quantization of periods
Do these forms have quantized periods?
In general, probably not. But invertible systems are special.
Relation with TQFT predicts that the WZW class is integral for families ofinvertible systems parameterized by Sd+2.
We gave a non-rigorous argument that this is indeed the case. Can bemade rigorous (N. Sopenko, last week).
For invertible bosonic systems in low dimensions (d < 4) expectquantization to hold for arbitrary cycles.
Need a more powerful approach to check this.I know how to assign a gerbeto a family of 1d gapped systems. Higher gerbes must be lurking aroundfor d > 1.
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 30 / 31
Some lessons
The space of gapped lattice systems has a complicated topology
In the invertible case, it is probably an infinite loop space
Coarse geometry in the style of John Roe plays an important role inthe study of gapped phases
Quantum statistical mechanics (study of dynamical systems based onlattice Hamiltonians with good locality properties) gives rise tointricate geometric, topoogical, and algebraic structures
Anton Kapustin (California Institute of Technology)From gapped phases of matter to Topological Quantum Field Theory and back againJune 22, 2020 31 / 31